master > master: updated comments and user input

README.md

@@ -18,4 +18,40 @@ See the [docs/*.ipynb](docs) files.
 **Note:** The Git provider _Gitea_ does not have a display format for ipynb-files.
 To properly view and run the notebooks, users require python v3.10.x and the jupyter module.
 To install the package requirements, call `just build`
-or `python3 -m pip install -r requirements.txt`.
+or `python3 -m pip install -r requirements.txt`.#
+
+## Usage ##
+
+To run the presentations:
+
+- Ensure you have a working copy of [**python 3.10.x**](https://www.python.org/downloads) on your system with rights to install packages.
+- Clone the repository.
+- Install the requirements. Navigate to the root of the repository and carry out:
+  ```bash
+  python3 -m pip install --disable-pip-version-check -r requirements.txt
+  # for Windows users:
+  py -3 -m pip install --disable-pip-version-check -r requirements.txt
+  ```
+- Start the desired notebooks using the [jupyter](https://jupyter.org) UI or run
+  ```bash
+  python3 -m jupyter notebook notebooks/xyz.ipynb
+  # for Windows users:
+  py -3 -m jupyter notebook notebooks/xyz.ipynb
+  ```
+  to run the notebook `docs/xyz.ipynb`.
+
+Alternatively we recommend the following:
+
+- Ensure you have a working copy of [**python 3.10.x**](https://www.python.org/downloads) on your system with rights to install packages.
+- Ensure you have **bash** (for Windows users see the [Git SCM](https://gitforwindows.org))
+  and the [**justfile**](https://github.com/casey/just#installation) tool
+  (Windows users may wish to install [Chocolatey](https://chocolatey.org/install) beforehand).
+- Clone the repository.
+- Navigate in a bash terminal to the root of the repository and make use of the following commands:
+  ```bash
+  # for setup:
+  just build
+  # to run the notebook docs/xyz.ipynb:
+  just run xyz
+  ```
+  (ensure that the extension is no included in the command!).

docs/examples_dilations.ipynb

@@ -30,9 +30,8 @@
     "import sys;\n",
     "\n",
     "# NOTE: need this to force jupyter to reload imports:\n",
-    "for key in list(sys.modules.keys()):\n",
-    "    if key.startswith('src.'):\n",
-    "        del sys.modules[key];\n",
+    "for key in filter(lambda key: key.startswith('src.'), list(sys.modules.keys())):\n",
+    "    del sys.modules[key];\n",
     "\n",
     "os.chdir(os.path.dirname(_dh[0]));\n",
     "sys.path.insert(0, os.getcwd());\n",
@@ -48,18 +47,12 @@
    "outputs": [],
    "source": [
     "# User input:\n",
-    "N = 4; # dimension of the Hilbert space.\n",
+    "N = 6; # dimension of the Hilbert space; must be divisible by 2 for this construction.\n",
     "d = 4;\n",
-    "\n",
-    "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
-    "# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n",
-    "alpha = 1/math.sqrt(d - 0.5);\n",
-    "\n",
-    "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
-    "# alpha = 1/math.sqrt(d - 1);\n",
-    "\n",
-    "# Otherwise:\n",
-    "# alpha = 1;"
+    "# force k-th order dissipation operators to be strictly postive for k < k0\n",
+    "# force k-th order dissipation operators to be non-positive for k >= k0\n",
+    "# one can choose 2 <= k0 <= d\n",
+    "k0 = d;"
    ]
   },
   {
@@ -69,6 +62,7 @@
    "outputs": [],
    "source": [
     "# create the generators `A_i` of the marginal semigroups `T_i`:`\n",
+    "alpha = 1/math.sqrt(k0 - 0.5);\n",
     "A = [\n",
     "    generate_semigroup_generator(\n",
     "        shape    = [N, N],\n",

src/examples_dilations.py

@@ -46,7 +46,7 @@ def generate_semigroup_generator(
     - `base`     - <int>      If `rational = True`, fixes the denominator of the rational numbers.
     - `alpha`    - <float>    Additional parameter to scale the D_i operators.
 
-    NOTE: in the paper `α = 1` was chosen. However one can choose any value in `(1/√d, \infty)`.
+    NOTE: One can choose any value of `α ∈ (1/√d, \infty)`.
     By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3
     one can force that the S_TK operators only fail to be positive when |K| > d-1.